Reducing u(t)+Int(0->t) to Differential Equation

In summary, to reduce the equation u(t)+Int(0->t) {[e^a(t-t1)]u(t1)}dt1 = k to a differential equation, first evaluate the equation at t=0. This shows that u(0) = 0. Then, differentiate the equation to get the differential equation using the Leibniz Integral Rule. For the initial condition, we know that the integral from 0 to 0 of any function is equal to 0.
  • #1
yukcream
59
0
How to reduce the equation: Show how the equation u(t)+Int( 0->t) {[e^a(t-t1)]u(t1)}dt1 = k can be reduced to a differential equation and obtain an intial condidtion for the equation.

Remarks: Int(0->t): integral from 0 to t !
 
Physics news on Phys.org
  • #2
Evaluate your equation at t=0. What does that say about u(0)?

Differentiate your equation to get your differential equation.
 
  • #3
arildno said:
Evaluate your equation at t=0. What does that say about u(0)?

Differentiate your equation to get your differential equation.

Yes I know I have to differential but if the diff. it with respect to t , what will happen to the second term on the left hand side?
 
  • #4
yukcream said:
Yes I know I have to differential but if the diff. it with respect to t , what will happen to the second term on the left hand side?
Use the Leibniz Integral Rule.
[tex]\frac{d}{dt}\int_{a(t)}^{b(t)}f(x,t)dx=\int_{a(t)}^{b(t)}
\partial_tf(x,t)dx+f(x,t)\partial_tx|_{x=a(t)}^{x=b(t)}[/tex]
[tex]\frac{d}{dt}\int_{a(t)}^{b(t)}f(x,t)dx=\int_{a(t)}^{b(t)}
\frac{\partial}{\partial t}f(x,t)dx+f(x,t){\partial x}{\partial t}\right|_{x=a(t)}^{x=b(t)}[/tex]
http://mathworld.wolfram.com/LeibnizIntegralRule.html
for the initial condition you should know that
[tex]\int_0^0 f(x) dx=0[/tex]
 
Last edited:

1. What is the purpose of reducing u(t)+Int(0->t) to a differential equation?

Reducing u(t)+Int(0->t) to a differential equation allows us to model and solve complex systems in a more simplified manner. This can help us better understand the behavior of the system and make predictions about its future outcomes.

2. How do you reduce u(t)+Int(0->t) to a differential equation?

To reduce u(t)+Int(0->t) to a differential equation, we need to use the Fundamental Theorem of Calculus. This theorem states that the derivative of the integral of a function is equal to the original function. By applying this theorem, we can rewrite the integral as a derivative, which can then be solved using traditional differential equations techniques.

3. What are some real-world applications of reducing u(t)+Int(0->t) to a differential equation?

Reducing u(t)+Int(0->t) to a differential equation is commonly used in physics, engineering, and economics to model and solve problems related to motion, heat transfer, and growth processes. For example, it can be used to predict the behavior of a falling object, the temperature distribution in a room, or the population growth of a species.

4. Are there any limitations to reducing u(t)+Int(0->t) to a differential equation?

While reducing u(t)+Int(0->t) to a differential equation can be a powerful tool, it does have some limitations. It may not be applicable to complex systems with nonlinear or time-dependent behavior. Additionally, it may not always provide an exact solution and may require additional simplifying assumptions.

5. Can the process of reducing u(t)+Int(0->t) to a differential equation be automated?

Yes, there are computer programs and software that can automatically reduce u(t)+Int(0->t) to a differential equation. These programs use algorithms and numerical methods to approximate the solution and can handle more complex and nonlinear systems. However, it is still important for a scientist to have a good understanding of the underlying principles and assumptions involved in the process.

Similar threads

I On vibrating string and differentiating infinite sum
    • Differential Equations
Replies
7
Views
391
I Verifying properties of Green's function
    • Differential Equations
Replies
1
Views
753
I Understanding relationship between heat equation & Green's function
    • Differential Equations
Replies
6
Views
2K
A Differential equation and Appell polynomials
    • Differential Equations
Replies
1
Views
1K
I Problem with integrating the differential equation more than once
    • Differential Equations
Replies
1
Views
1K
I Determining equilibrium solutions to differential equations
    • Differential Equations
Replies
16
Views
894
I Solution of delayed forcing function
    • Differential Equations
Replies
5
Views
1K
I On sub and super solutions: Teschl and others
    • Differential Equations
Replies
5
Views
654
I On error estimates of approximate solutions
    • Differential Equations
Replies
1
Views
666
I Fundamental matrix of a second order 2x2 system of ODEs
    • Differential Equations
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top